With the increase of \(J_{1}\) , the cubic-root system could also undergo
a series of topological phase transitions, with each of them being
accompanied by the increase of either \(n_{0}\)  and \(n_{2\pi /3}\)  or \(n_{\pi /3}\)  and \(n_{\pi }\) 
by two. We can thus obtain arbitrarily many \(\pi /3\)  and \(2\pi /3\) 
degenerate edge modes by tuning the single driving parameter \(J_{1}\) 
even in the presence of NHSE. Since it has been demonstrated that
Floquet edge states could be utilized to construct boundary discrete time crystals
(DTCs) [1]}, [2]}, we expect the emergence of unique non-Hermitian
Floquet boundary DTCs through the superposition of \((\pm \pi /3,\pm 2\pi /3)\)  edge modes
and other edge states in the cubic-root FTIs.
